The private pilot reported that he was receiving commercial instruction in an airplane with retractable landing gear during a cross country flight to an unfamiliar airport. He added that while he turned the airplane from the downwind leg to the base leg of the traffic pattern for the runway, he did not perform the checklist, and the flight instructor did not verity that the landing gear were extended. The airplane touched down on the runway with the landing gear retracted. The airplane sustained substantial damage to the external longerons on the lower fuselage. The pilot reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation. The pilot did not submit the National Transportation Safety Board Pilot/Operator Aircraft Accident/Incident Report Form 6120.1.